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How to solve interest problem without using a crazy binomial expansion?
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Problem 9.9
Eric deposits X into a savings account at time 0, which pays interest at
a nominal rate of i, compounded semiannually. Mike deposits 2X into a
different savings account at time 0, which pays simple interest at an annual
rate of i. Eric and Mike earn the same amount of interest during the last 6
months of the 8th year. Calculate i.
Hello everyone, I'm doing a practice problem from Finan's actuarial exam FM textbook, and there is an equation I came up with that comes up with the right answer, but I have to use a graphing calculator to solve it.
You can't use a graphing calculator on the actuary exams, so I have to find out a way to solve this without getting a headache with the binomial theorem.
First, I get this: $$A_E(8) - A_E(7.5) = A_M(8) - A_M(7.5)$$
Simplifying all that mess, I get this monster:
$$(1+ frac i2)^16 - (1+ frac i2)^15 = i$$
(I simplified all the i's because of simple interest)
Putting this beast in my calculator, I get the answer: $$i=.094588$$ ,which is the correct answer. Maybe you can figure out an easier way to solve this, but I would also like to know if there is a way to solve it this way without the use of a graphing calculator.
I'm thinking that this could possibly be solved by some sort of series expansion.
statistics finance binomial-theorem actuarial-science
asked Jul 31 at 18:00
Sweet Genius
82
add a comment |Â
up vote
1
down vote
favorite
Problem 9.9
Eric deposits X into a savings account at time 0, which pays interest at
a nominal rate of i, compounded semiannually. Mike deposits 2X into a
different savings account at time 0, which pays simple interest at an annual
rate of i. Eric and Mike earn the same amount of interest during the last 6
months of the 8th year. Calculate i.
Hello everyone, I'm doing a practice problem from Finan's actuarial exam FM textbook, and there is an equation I came up with that comes up with the right answer, but I have to use a graphing calculator to solve it.
You can't use a graphing calculator on the actuary exams, so I have to find out a way to solve this without getting a headache with the binomial theorem.
First, I get this: $$A_E(8) - A_E(7.5) = A_M(8) - A_M(7.5)$$
Simplifying all that mess, I get this monster:
$$(1+ frac i2)^16 - (1+ frac i2)^15 = i$$
(I simplified all the i's because of simple interest)
Putting this beast in my calculator, I get the answer: $$i=.094588$$ ,which is the correct answer. Maybe you can figure out an easier way to solve this, but I would also like to know if there is a way to solve it this way without the use of a graphing calculator.
I'm thinking that this could possibly be solved by some sort of series expansion.
statistics finance binomial-theorem actuarial-science
asked Jul 31 at 18:00
Sweet Genius
82
I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Problem 9.9
Eric deposits X into a savings account at time 0, which pays interest at
a nominal rate of i, compounded semiannually. Mike deposits 2X into a
different savings account at time 0, which pays simple interest at an annual
rate of i. Eric and Mike earn the same amount of interest during the last 6
months of the 8th year. Calculate i.
Hello everyone, I'm doing a practice problem from Finan's actuarial exam FM textbook, and there is an equation I came up with that comes up with the right answer, but I have to use a graphing calculator to solve it.
You can't use a graphing calculator on the actuary exams, so I have to find out a way to solve this without getting a headache with the binomial theorem.
First, I get this: $$A_E(8) - A_E(7.5) = A_M(8) - A_M(7.5)$$
Simplifying all that mess, I get this monster:
$$(1+ frac i2)^16 - (1+ frac i2)^15 = i$$
(I simplified all the i's because of simple interest)
Putting this beast in my calculator, I get the answer: $$i=.094588$$ ,which is the correct answer. Maybe you can figure out an easier way to solve this, but I would also like to know if there is a way to solve it this way without the use of a graphing calculator.
I'm thinking that this could possibly be solved by some sort of series expansion.
statistics finance binomial-theorem actuarial-science
asked Jul 31 at 18:00
Sweet Genius
82
Problem 9.9
Eric deposits X into a savings account at time 0, which pays interest at
a nominal rate of i, compounded semiannually. Mike deposits 2X into a
different savings account at time 0, which pays simple interest at an annual
rate of i. Eric and Mike earn the same amount of interest during the last 6
months of the 8th year. Calculate i.
Hello everyone, I'm doing a practice problem from Finan's actuarial exam FM textbook, and there is an equation I came up with that comes up with the right answer, but I have to use a graphing calculator to solve it.
You can't use a graphing calculator on the actuary exams, so I have to find out a way to solve this without getting a headache with the binomial theorem.
First, I get this: $$A_E(8) - A_E(7.5) = A_M(8) - A_M(7.5)$$
Simplifying all that mess, I get this monster:
$$(1+ frac i2)^16 - (1+ frac i2)^15 = i$$
(I simplified all the i's because of simple interest)
Putting this beast in my calculator, I get the answer: $$i=.094588$$ ,which is the correct answer. Maybe you can figure out an easier way to solve this, but I would also like to know if there is a way to solve it this way without the use of a graphing calculator.
I'm thinking that this could possibly be solved by some sort of series expansion.
statistics finance binomial-theorem actuarial-science
asked Jul 31 at 18:00
Sweet Genius
82
asked Jul 31 at 18:00
Sweet Genius
82
asked Jul 31 at 18:00
Sweet Genius
82
asked Jul 31 at 18:00
Sweet Genius
82
82
I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05
add a comment |Â
I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05
I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05
I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
0
down vote
accepted
After $7.5$ years, Eric has $X(1+frac i2)^15$ capital and thus makes an additional $X(1+frac i2)^15frac i2$ in the final six months. In the final year, Mike makes $2Xi$ from simple interest on his initial $2X$ investment. Thus in the final six months, he makes... well, he still just makes $2Xi$, delivered in December. However, if we assume that, we get the wrong answer, so it seems the question means for you to interpret "money made in the last half year" to mean "half the amount made in the full year", so we say he makes $Xi$ in the last six months. Thus we have:
$$X(1+frac i2)^15frac i2=Xi$$
this simplifies to
$$(1+frac i2)^15=2$$
and so $i=2(sqrt[15]2-1)$, as in the other answer.
answered Jul 31 at 20:12
Jack M
16.9k33473
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
add a comment |Â
up vote
1
down vote
$$left(1+fraci2right)^16-left(1+fraci2right)^15~=~i$$
First in both terms of the L.H.S. contain the $15^th$ power of $left(1+fraci2right)$.
So lets rearrange this a little bit
$$beginalign
left(1+fraci2right)^16-left(1+fraci2right)^15~&=~i\
left(1+fraci2right)^15left(left(1+fraci2-1right)right)~&=~i\
left(1+fraci2right)^15left(fraci2right)~&=~i\
endalign$$
Now dividing both sides by $i$ and taking the $15^th$ root yields to
$$beginalign
left(1+fraci2right)^15left(frac12right)~&=~1\
left(1+fraci2right)^15~&=~2\
left(1+fraci2right)~&=~sqrt[15]2\
i~&=2cdot(sqrt[15]2-1)
endalign$$
which is approximately $i=0.094588245641$, your given solution.
answered Jul 31 at 18:10
mrtaurho
660117
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
After $7.5$ years, Eric has $X(1+frac i2)^15$ capital and thus makes an additional $X(1+frac i2)^15frac i2$ in the final six months. In the final year, Mike makes $2Xi$ from simple interest on his initial $2X$ investment. Thus in the final six months, he makes... well, he still just makes $2Xi$, delivered in December. However, if we assume that, we get the wrong answer, so it seems the question means for you to interpret "money made in the last half year" to mean "half the amount made in the full year", so we say he makes $Xi$ in the last six months. Thus we have:
$$X(1+frac i2)^15frac i2=Xi$$
this simplifies to
$$(1+frac i2)^15=2$$
and so $i=2(sqrt[15]2-1)$, as in the other answer.
answered Jul 31 at 20:12
Jack M
16.9k33473
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
add a comment |Â
up vote
0
down vote
accepted
After $7.5$ years, Eric has $X(1+frac i2)^15$ capital and thus makes an additional $X(1+frac i2)^15frac i2$ in the final six months. In the final year, Mike makes $2Xi$ from simple interest on his initial $2X$ investment. Thus in the final six months, he makes... well, he still just makes $2Xi$, delivered in December. However, if we assume that, we get the wrong answer, so it seems the question means for you to interpret "money made in the last half year" to mean "half the amount made in the full year", so we say he makes $Xi$ in the last six months. Thus we have:
$$X(1+frac i2)^15frac i2=Xi$$
this simplifies to
$$(1+frac i2)^15=2$$
and so $i=2(sqrt[15]2-1)$, as in the other answer.
answered Jul 31 at 20:12
Jack M
16.9k33473
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
After $7.5$ years, Eric has $X(1+frac i2)^15$ capital and thus makes an additional $X(1+frac i2)^15frac i2$ in the final six months. In the final year, Mike makes $2Xi$ from simple interest on his initial $2X$ investment. Thus in the final six months, he makes... well, he still just makes $2Xi$, delivered in December. However, if we assume that, we get the wrong answer, so it seems the question means for you to interpret "money made in the last half year" to mean "half the amount made in the full year", so we say he makes $Xi$ in the last six months. Thus we have:
$$X(1+frac i2)^15frac i2=Xi$$
this simplifies to
$$(1+frac i2)^15=2$$
and so $i=2(sqrt[15]2-1)$, as in the other answer.
answered Jul 31 at 20:12
Jack M
16.9k33473
After $7.5$ years, Eric has $X(1+frac i2)^15$ capital and thus makes an additional $X(1+frac i2)^15frac i2$ in the final six months. In the final year, Mike makes $2Xi$ from simple interest on his initial $2X$ investment. Thus in the final six months, he makes... well, he still just makes $2Xi$, delivered in December. However, if we assume that, we get the wrong answer, so it seems the question means for you to interpret "money made in the last half year" to mean "half the amount made in the full year", so we say he makes $Xi$ in the last six months. Thus we have:
$$X(1+frac i2)^15frac i2=Xi$$
this simplifies to
$$(1+frac i2)^15=2$$
and so $i=2(sqrt[15]2-1)$, as in the other answer.
answered Jul 31 at 20:12
Jack M
16.9k33473
answered Jul 31 at 20:12
Jack M
16.9k33473
answered Jul 31 at 20:12
Jack M
16.9k33473
answered Jul 31 at 20:12
Jack M
16.9k33473
16.9k33473
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
add a comment |Â
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
Another way of looking at it; thanks much!
– Sweet Genius
Jul 31 at 20:56
add a comment |Â
up vote
1
down vote
$$left(1+fraci2right)^16-left(1+fraci2right)^15~=~i$$
First in both terms of the L.H.S. contain the $15^th$ power of $left(1+fraci2right)$.
So lets rearrange this a little bit
$$beginalign
left(1+fraci2right)^16-left(1+fraci2right)^15~&=~i\
left(1+fraci2right)^15left(left(1+fraci2-1right)right)~&=~i\
left(1+fraci2right)^15left(fraci2right)~&=~i\
endalign$$
Now dividing both sides by $i$ and taking the $15^th$ root yields to
$$beginalign
left(1+fraci2right)^15left(frac12right)~&=~1\
left(1+fraci2right)^15~&=~2\
left(1+fraci2right)~&=~sqrt[15]2\
i~&=2cdot(sqrt[15]2-1)
endalign$$
which is approximately $i=0.094588245641$, your given solution.
answered Jul 31 at 18:10
mrtaurho
660117
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
add a comment |Â
up vote
1
down vote
$$left(1+fraci2right)^16-left(1+fraci2right)^15~=~i$$
First in both terms of the L.H.S. contain the $15^th$ power of $left(1+fraci2right)$.
So lets rearrange this a little bit
$$beginalign
left(1+fraci2right)^16-left(1+fraci2right)^15~&=~i\
left(1+fraci2right)^15left(left(1+fraci2-1right)right)~&=~i\
left(1+fraci2right)^15left(fraci2right)~&=~i\
endalign$$
Now dividing both sides by $i$ and taking the $15^th$ root yields to
$$beginalign
left(1+fraci2right)^15left(frac12right)~&=~1\
left(1+fraci2right)^15~&=~2\
left(1+fraci2right)~&=~sqrt[15]2\
i~&=2cdot(sqrt[15]2-1)
endalign$$
which is approximately $i=0.094588245641$, your given solution.
answered Jul 31 at 18:10
mrtaurho
660117
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
add a comment |Â
up vote
1
down vote
up vote
1
down vote
$$left(1+fraci2right)^16-left(1+fraci2right)^15~=~i$$
First in both terms of the L.H.S. contain the $15^th$ power of $left(1+fraci2right)$.
So lets rearrange this a little bit
$$beginalign
left(1+fraci2right)^16-left(1+fraci2right)^15~&=~i\
left(1+fraci2right)^15left(left(1+fraci2-1right)right)~&=~i\
left(1+fraci2right)^15left(fraci2right)~&=~i\
endalign$$
Now dividing both sides by $i$ and taking the $15^th$ root yields to
$$beginalign
left(1+fraci2right)^15left(frac12right)~&=~1\
left(1+fraci2right)^15~&=~2\
left(1+fraci2right)~&=~sqrt[15]2\
i~&=2cdot(sqrt[15]2-1)
endalign$$
which is approximately $i=0.094588245641$, your given solution.
answered Jul 31 at 18:10
mrtaurho
660117
$$left(1+fraci2right)^16-left(1+fraci2right)^15~=~i$$
First in both terms of the L.H.S. contain the $15^th$ power of $left(1+fraci2right)$.
So lets rearrange this a little bit
$$beginalign
left(1+fraci2right)^16-left(1+fraci2right)^15~&=~i\
left(1+fraci2right)^15left(left(1+fraci2-1right)right)~&=~i\
left(1+fraci2right)^15left(fraci2right)~&=~i\
endalign$$
Now dividing both sides by $i$ and taking the $15^th$ root yields to
$$beginalign
left(1+fraci2right)^15left(frac12right)~&=~1\
left(1+fraci2right)^15~&=~2\
left(1+fraci2right)~&=~sqrt[15]2\
i~&=2cdot(sqrt[15]2-1)
endalign$$
which is approximately $i=0.094588245641$, your given solution.
answered Jul 31 at 18:10
mrtaurho
660117
edited Jul 31 at 18:18
answered Jul 31 at 18:10
mrtaurho
660117
answered Jul 31 at 18:10
mrtaurho
660117
answered Jul 31 at 18:10
mrtaurho
660117
660117
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
add a comment |Â
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
Wow, that easy lol? I don't know why I didn't think of that. Thank you very much!
– Sweet Genius
Jul 31 at 18:20
add a comment |Â
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I don't know why the binomial theorem would be involved. It sounds like a simple set of simultaneous equations that can be solved by many methods using a simple calculator if needed. Most of the work is on paper. The calculator just saves doing arithmetic by hand.
– poetasis
Jul 31 at 18:05